A model typically used to represent simplest forms of granular systems consisted of a one dimensional chain of spherical beads regulated by Hertzian contact interaction potentials. However, a general wave dynamic theory supporting compact solitary waves was derived for structured homogeneous materials showing a highly nonlinear force (F)-displacement (δ) response dictated by the intrinsically nonlinear potential of interaction between its fundamental components. This general nonlinear spring-type contact relation can be expressed as:F≅Aδn,  (1)where n is a nonlinear exponent of the contact interaction (n>1) of a fundamental component and A is parameter for the material. For Hertzian systems, such as those consisting of a chain of spherical beads, the n exponent of interaction is equal to 1.5.
Within the present disclosure, “granular matter” is defined as an aggregate of “particles” in elastic contact with each other, preferably in linear or network shaped arrangements. In addition to the nonlinear contact interaction and the particle's geometry, another unique feature of the granular state is provided by a so-called zero tensile strength, which introduces additional nonlinearity (asymmetric potential) to the overall response. Consequently, in the absence of static pre-compression on the system, the linear range becomes negligible in the interaction of forces between neighboring particles, thus leading to materials with a characteristic sound speed equal to zero in its uncompressed state (c0=0), known as a “sonic vacuum”. This highly nonlinear wave theory supports, in particular, a new type of compact highly tunable solitary waves that have been experimentally and numerically observed in several works for the case of one-dimensional Hertzian granular systems.